P vs T Assignment: Radiated Power as a Function of Temperature
P vs T Assignment: Radiated Power as a Function of Temperature
For an ideal blackbody in thermal equilibrium, theory predicts that the power (P) radiated by the object varies as the 4th power of its temperature, i.e.
[latex]P \propto T^4[/latex]
The filament of a light bulb in thermal equilibrium loses most of its energy to radiation and so might be expected to behave in a similar way. The temperature of a tungsten filament can be measured through its resistance, π , by
| [latex]T = 130.7 + 189.8 \left( \frac{R}{R_o}\right) - 1.327 \left( \frac{R}{R_o}\right)^2 \ \ [^\circ K][/latex] | (1) |
where π π is the room temperature resistance of the filament. By adding an ammeter and a voltmeter to the bulbβs circuit, you can measure both the resistance of the bulb ([latex]π = πβπΌ[/latex]) and the power expended ([latex]π = πΌπ[/latex]).
Part 1: Determination of Ro
a) Summarize the data from Measurement Set #1 (measured voltage, measured current, and calculated resistance) with uncertainties for each, and present it in a table in best form. The specs for the orange and blue meters are available here.
b) Plot a graph (with uncertainties) showing your calculated π as a function of voltage. The graph should be suitably sized and scaled, and show data points with uncertainties.
c) Use the graph to provide a realistic estimate of π π, with uncertainty.
Part 2: Power vs Temperature measurement
a) Summarize the data from Measurement Set #2 (measured voltage and current, calculated resistance, temperature, and power), with uncertainties, and present it in a table in best form.
b) Plot a graph (with uncertainties) showing your calculated π as a function of π. The graph should be suitably sized and scaled, and show data points with uncertainties.
Part 3: Ln-Ln AnalysisΒ
a) Create a separate table of Ln(T) and Ln(P), with uncertainties.
b) Graph the Ln-Ln data. Data points should be shown with uncertainties. Your graph should be suitably sized and scaled so that all relevant detail is visible. A trendline and descriptive equation should appear on the graph.
c) Estimate the exponent, with uncertainty, for the P vs T function. Comment on your results.
Propagation of Uncertainties
The most common mistake in propagation occurs during multiplication or division. Take as an example the calculation of the uncertainty on the power ([latex]π = πΌπ[/latex]) from a measured current, πΌ Β± βπΌ, and a measured voltage, π Β± βπ . You are most likely to calculate the relative uncertainty correctly as
| [latex]\frac{\Delta P}{P} = \sqrt{ \left( \frac{\Delta V}{V}\right)^2 + \left(\frac{\Delta I}{I}\right) ^2 }[/latex] | Β Β (2) |
But… this is not the absolute uncertainty on π. (Among other things, this quantity is dimensionless β it has no units.) To complete the calculation, you need to multiply by π, i.e.
| [latex]\Delta P = P \sqrt{\left( \frac{\Delta V}{V}\right)^2 + \left( \frac{\Delta I}{I}\right) ^2 }[/latex] | (3) |
In the temperature calculation, the uncertainty on the temperature should be calculated using the function argument
i.e. if π¦ = π(π₯) , then
| [latex]\Delta y = \frac{dy}{dx}\Delta x[/latex] | Β (4) |
where βπ₯ is the uncertainty on π₯. In your case, the temperature is a function of two independent variables, R and Ro, so the uncertainty should be calculated as
| [latex]\Delta T = \sqrt {\left( \frac{\delta T}{\delta R}\Delta R \right)^2+\left( \frac{\delta T}{\delta R_o}\Delta R_o \right)^2 }[/latex] | (5) |
Provide a sample of your uncertainty calculations.